Limiting behavior of random permanents
Given a matrix \(X=[X_{i,j}]\), \(i=1,\dots, m\), \(j=1,\dots, n,\) where \(X_{i,j}\) are i.i.d. r.v.'s with \(EX_{1,1}=\mu\neq 0\) and \(\operatorname {var} X_{1,1}=\sigma^2\), let \(\operatorname {Per} (X)\) be the permanent of \(X\) and set \(S_{m,n}= \operatorname {Per} (X)[\nu m\mu!\mu^m]^{-1}\), \(\gamma=\sigma/\mu.\) If \(m,n\to\infty\) so that \(m/n\to\lambda >0\), then \(S_{m,n}@>D>> \exp(\sqrt{\lambda}\gamma N-\lambda\gamma^2/2),\) \(N\) being a standard normal r.v., and if \(\lambda =0\), then \(\gamma^{-1}\sqrt{n/m}(S_{m,n}-1) @>D>> N.\) The proofs are based on analysis of variance of random permanents and involve related asymptotic properties of elementary symmetric polynomials in \(X_{i,j}.\) The results of \textit{V. S. Korolyuk} and \textit{Yu. V. Borovskikh} [in: Probability theory and mathematical statistics, 176-187 (1992)] and \textit{A. J. van Es} and \textit{R. Helmers} [Probab. Theory Relat. Fields 80, No. 1, 21-35 (1988; Zbl 0638.60028)] are employed.
scientific article; zbMATH DE number 1367949